Asymmetries of anti-triplet charmed baryon decays

We analyze the decay processes of ${\bf B}_c \to {\bf B}_n M$ with the $SU(3)_F$ flavor symmetry and spin-dependent amplitudes, where ${\bf B}_c({\bf B}_n)$ and $M$ are the anti-triplet charmed (octet) baryon and nonet meson states, respectively. In the $SU(3)_F$ approach, it is the first time that the decay rates and asymmetries are fully and systematically studied without neglecting the ${\cal O}(\overline{15})$ contributions of the color anti-symmetric parts in the effective Hamiltonian. Our results of the up-down asymmetries based on $SU(3)_F$ are quite different from the previous theoretical values in the literature. In particular, we find that $ \alpha(\Lambda_c^+\to \Xi^0 K^+)_{SU(3)} = 0.94^{+0.06}_{-0.11}$, which is consistent with the recent experimental data of $0.77\pm0.78$ by the BESIII Collaboration, but predicted to be zero in the literature. We also examine the $K_S^0-K_L^0$ asymmetries between the decays of ${\bf B}_c \to {\bf B}_n K_S^0$ and ${\bf B}_c \to {\bf B}_n K_L^0$ with both Cabibbo-allowed and doubly Cabibbo-suppressed transitions.


I. INTRODUCTION
Recently, the Belle collaboration has measured the absolute branching ratio of Λ + c → pK − π + with high precision [1], resulting in the world average value of B(Λ + c → pK − π + ) = (6.23 ± 0.33)%, given by the Particle Data Group (PDG) [2]. This decay mode is the socalled golden channel as most of Λ + c decay branching fractions are presented relative to it. Subsequently, this golden mode, along with many other Λ + c ones, has also been observed by the BESIII Collaboration [3][4][5][6][7][8][9][10][11][12] with Λ + cΛ − c pairs, produced by e + e − collisions at a center-ofmass energy of √ s = 4.6 GeV, having a uniquely clean background to study the anti-triplet charmed baryon state of Λ + c . In particular, the decay of Λ + c → Σ + η ′ has been seen for the first time with η ′ in the final states for the charmed baryon decays [12]. In addition, the absolute decay branching fraction of Ξ 0 c → Ξ − π + , which involves the anti-triplet charmed baryon state of Ξ 0 c , has also been measured by the Belle collaboration [13]. Clearly, a new experimental physics era for charmed baryons has started.
For the two-body charmed baryon decay of B c → B n M, with B c (B n ) and M the antitriplet charmed (octet) baryon and nonet meson states, respectively, beside its decay branching fraction, there exits another interesting physical observable, the up-down asymmetry α, which is related to the longitudinal polarization of B n . Currently, there are three experi-mental measurements of the up-down asymmetries in the charmed baryon decays [2], along with the recent one by BESIII [11], given by which has been suggested to be approximately zero in the previous theoretical studies with the dynamical models [26-31, 33, 34] as well as the SU(3) F approach [16]. However, the up-down asymmetries in B c → B n M were not discussed in the previous studies with SU(3) F in Refs. [17][18][19][20][21][22][23][24][25].
In addition, it has been noticed that the physical Cabibble-allowed dominated decay processes of B c → B n K 0 L and B c → B n K 0 S are the same when the doubly Cabibbo-suppressed contributions are taking to be zero [19]. However, in some of these processes, the doubly Cabibbo-suppressed transitions are not negligible, which can be examined by defining the K 0 S − K 0 L asymmetries between the K 0 S and K 0 L modes [19] to track the interferences. In this work, we will systematically analyze the decay processes of B c → B n M with the SU(3) F symmetry with all operators under SU(3) F . We will also include the effect of the η − η ′ mixing. There are two different ways to link the amplitudes among the processes by SU(3) F . The first one is a purely mathematical consideration. By imposing the SU (3) group, we are able to write down the amplitude by tensor contractions. The second one is the diagrammatic approach, in which one draws down all the possible diagrams for the decay process with ascertaining that the amplitude from each diagram shall be the same by interchanging up, down and strange quarks. Both ways have their own advantages. The tensor method is easier to cooperate with the other symmetry and it allows us to estimate the order of the contribution from the amplitude with the Wilson coefficients. Explicitly, it could cooperate with the SU(3) color symmetry and take account of the strange quark mass as the source of the SU(3) F symmetry breaking [15,22]. On the other hand, the diagrammatic approach can distinguish the factorizable and non-factorizable amplitudes [32]. The close relations between the two methods have been examined in Ref. [37]. In Ref. [23], it has been proved to be useful if one combines both methods. This paper is organized as follows. In Sec. II, we give the formalism for the two-body charmed baryon decays of B c → B n M, in which we first write the decay amplitudes in terms of parity conserved and violated parts under the SU(3) F flavor symmetry, and then display the decay rates and asymmetries. In Sec. III, we show our numerical results and present discussions. We conclude in Sec. IV. In Appendix A, we list the all decay amplitudes of the anti triplet baryon states in terms of the SU(3) F parameters. We give the definitions of the up-down and longitudinal polarization asymmetries in Appendix B.

II. FORMALISM
To study the two-body decays of the anti-triplet charmed baryon (B c ) to octet baryon respectively. In general, we write the spin-dependent amplitude of B c → B n M as where A and B are the s-wave and p-wave amplitudes, corresponding to the parity violating and conserving ones, and u Bn,c are the baryon Dirac spinors, respectively. From Eqs. (3) and (6), we can decompose A and B in terms of the tensor forms under SU(3) F as meson state M is directly given by the weak interaction alone as demonstrated in Ref. [23].
As a result, in our calculations we will neglect the terms associated with a 0 , a 4 , a 5 and a 7 The decay angular distribution of the directionp where P Bn is the polarization vector of B n with the longitudinal component being P Bn = α, θ is the angle between P Bn andp Bn and α is the so-called up-down asymmetry parameter, given by with E Bn and p Bn the energy and three momentum of B n . The definitions of the up-down and longitudinal asymmetries can be found in Appendix B. Consequently, we obtain the decay rate as To extract the doubly Cabibbo-suppressed contributions in the Cabibbo-allowed dominating
It is worth to take a closer look on the parameters in Eq. (14). As mentioned early, H (15) only contributes to the factorization amplitudes, which can be parametrized only in terms of a 6 and b 6 terms, corresponding to the vector and axial-vector currents in the baryonic matrix elements, respectively. Our result of b 6 ≫ a 6 in Eq. (14) suggests that the axial-vector part of the factorization contribution is much larger that the vector one. This can be understood as follows. In the decay of B c → B n M, the pseudoscalar meson part of the factorization approach is given by where f M is the meson decay constant, while q µ is the four-momentum of M, which is also equal to the four-momentum difference between the initial and final baryons of B c and B n .
Consequently, we get that where q stands for the light quarks. In the case of the SU(4) flavor symmetry, in which the charm quark is also treated as q, Eq. (17) is automatically satisfied as the right-handed part is zero. It is clear that the inequality in Eq. (17) depends on the parameters a 6 and b 6 , which are not quite determined yet, particularly a 6 . In fact, from Table IX in Appendix A, we have that in which a 2 and a 3 get almost canceled out each other, resulting in that it could be dominated by the a 6 terms. In this case, the experimental search for the up-down asymmetry as well as the future measurement on the branching ration of Λ + c → pπ 0 will be helpful to obtain the precise value of a 6 .
In Tables II, III and IV, we list our predictions of the branching ratios and up-down asymmetries for the Cabibbo-allowed, singly Cabibbo-suppressed and doubly Cabibbo-suppressed decays, respectively. In the tables, we have also presented the values of A and B, which are useful to understand the up-down asymmetries as well as the comparisons with those given by specific theoretical models. We note that some of our results for the up-down asymmetries have been discussed for the first time in the literature, while the decay branching ratios are almost the same as those in Refs. [17][18][19][20][21][22][23]. In particular, we find that B(Λ + c → pπ 0 ) = (1.2±1.2)×10 −4 , which is consistent with our previous value of (1.3±0.7)×10 −4 in Ref. [23] and 0.8 × 10 −4 calculated by the pole model with current algebra in Ref. [36] as well as the current experimental upper limit of 2.7 × 10 −4 [2]. In addition, the decay branching ratio for the related Cabibbo-suppressed mode of Λ + c → nπ + is predicted to be (8.5 ± 1.9) × 10 −4 , in comparison with (6.1 ± 2.0) × 10 −4 in Ref. [23] and 2.7 × 10 −4 in Ref. [36]. We remark that most of the predictions in the present work with the spin-dependent amplitudes have small uncertainties comparing to those of our previous study with SU(3) F in Ref. [23] except the decay of Λ + c → pπ 0 due to the cancellation effect as well as the correlations in Eq. (15). To compare our predictions of the up-down asymmetries with those in the literature, we summarize the values of α for the Cabibbo-allowed and singly Cabibbo-suppressed decays   Zen are based on the pole models by Xu and Kamal (XK) [27], Cheng and Tseng (CT) [29] and Zenczykowski (Zen) [31], SV1, CT ′ , UVK (′) and CKX are related to the considerations of current algebra by Sharma and Verma (SV1) [34], Cheng and Tseng (CT) [29], Uppal, Verma and Khanna (UVK) without (with) the baryon wave function scale variation [30] and Cheng, Kang and Xu (CKX) [36], and SV2 (′) represent the results with SU(3) F by Sharma and Verma with two different signs of B(Λ + c → Ξ 0 K + ) [16], respectively. As seen in Table V, our results of the up-down asymmetries are quite different from those in the literature [16, 26-31, 33, 34]. In particular, it is interesting to see that we predict that which is consistent with the current experimental data of 0.77 ± 0.78 in Eq. (1) [11], but    that this asymmetry is approximately zero in dynamical models [26-31, 33, 34], while the authors in Ref. [16] have also taken it to be zero as a data input when the SU(3) F symmetry is imposed. In our fit, the value in Eq. (1) has not been included as an input in order to see its value based on the SU(3) F approach. Since the error of our predicted result in Eq. (19) is small, we are confident that α(Λ + c → Ξ 0 K + ) should be much lager than zero and close to one. To clarify this issue, a further precision measurement on this asymmetry is highly recommended.
In addition, due to the vanishing contributions to the decays from the a 4 , a 5 , a 7 and a ′ 0 terms of O(15), we get leading to the fitted values of as given in Table III. Note that the decay branching ratio of Λ + c → Σ 0 K + has been measured to be (5.2 ± 0.8) × 10 −4 [2], which agrees with with that in Eq. (21). Future measurements on Λ + c → Σ + K 0 S and Λ + c → Σ + K 0 L are important as they can tell us if Eqs. (20) and (21), which can also be derived through the isospin symmetry, are right or wrong.
We now concentrate on the decay processes of B c → B n K 0 L and B c → B n K 0 S , which involve both Cabibbo-allowed and doubly suppressed transitions, as shown in Table VII. If we ignore the later contributions associated with sin 2 θ c , B(B c → B n K 0 S ) = B(B c → B n K 0 L ). Clearly, the K 0 S − K 0 L asymmetry depends on the doubly Cabibbo-suppressed parts of the decays. As shown in Table VII, the central values for the first three asymmetries are predicted to be around 10% or more, which are consistent with those in Ref. [19]. For Ξ 0 c → Σ 0 K 0 S /K 0 L , the up-down asymmetry of R K 0 S,L (Ξ 0 c → Σ 0 ) has different sign, indicating that the effect of the doublyCabibbo-suppressed transition is not ignorable in these decay processes. Explicitly, we find out that B(Ξ 0 c → Λ 0 K 0 L ) can be a little larger than B(Ξ 0 c → Λ 0 K 0 S ), in which the K 0 S − K 0 L asymmetry is predicted to be −(4.3 ± 0.3)% with a tiny uncertainty, which agrees well with −(3.7 ± 0.4)% in Ref. [19]. i , allow us to examine the longitudinal polarization of P Bn , which is related to the up-down asymmetry of α. We have obtained a good χ fit for the ten SU(3) F parameters in Eq. (14) from the all possible contributions of O(6) and O(15) with 16 data points in Table I in the SU(3) F approach, in which all experimental data for the decay branching ratios and up-down asymmetries can be explained. Consequently, we have systematically predicted all decay branching ratios and up-down asymmetries of the Cabibbo-allowed, singly Cabibbosuppressed and doubly Cabibbo-suppressed charmed baryon decays. In particular, our results of B(Ξ 0 c → Ξ − π + ) = (2.21 ± 0.14) × 10 −2 and α(Ξ 0 c → Ξ − π + ) = −0.98 +0.07 −0.02 are consistent with the data of (1.80 ± 0.52) × 10 −2 [13] and −0.6 ± 0.4 [2], respectively. We have also found that B(Λ + c → pπ 0 ) = (1.2 ± 1.2) × 10 −4 , which is consistent with the current experimental upper limit of 2.7 × 10 −4 [2]. In addition, we have gotten that B(Λ + c → Σ 0 K + , Σ + K 0 S , Σ + K 0 L ) = (5.4 ± 0.7) × 10 −4 and α(Λ + c → Σ 0 K + , Σ + K 0 S , Σ + K 0 L ) = −1.00 +0.06 −0.00 , which are also guaranteed by the isospin symmetry.
We have shown in Table V that our predictions of the up-down asymmetries are quite different from the theoretical values in the literature for most of the decay modes. In particular, we have found that α(Λ + c → Ξ 0 K + ) SU (3) = 0.94 +0.06 −0.11 in Eq. (19), which is consistent with the current experimental data of 0.77 ± 0.78 in Eq. (1) [11], but much larger than zero predicted in the literature. A future precision measurement on this asymmetry is clearly very important as our prediction based on SU(3) F is close to one with a small uncertainty, which can be viewed as a benchmark for the SU(3) F approach.
We have also explored the K 0 S − K 0 L asymmetries in the decays of B c → B n K 0 L /K 0 S with both Cabibbo-allowed and doubly Cabibbo-suppressed transitions. The asymmetries depend strongly on the contributions from the doubly Cabibbo-suppressed contributions. Clearly, the measurements of these asymmetries are good tests for the doubly Cabibbo-suppressed transitions.
In conclusion, we give a systematic consideration of the up-down asymmetries in the two-body charmed baryon decays of B c → B n M as well as the K 0 S − K 0 L asymmetries in the decays of B c → B n K 0 L /K 0 S in the SU(3) F approach. Some of our predictions based on SU(3) F are different from those in the dynamical models, can be tested by the experiments at BESIII and Belle.

Appendix A: Irreducible Amplitudes
In this Appendix, we provide the irreducible amplitudes A Bc→BnM from Eq. (7) which is equal to the longitudinal polarization asymmetry, i.e. P Bn = α.